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Calculus 2413 Ch 3 Section 1 Slope, Tangent Lines, and Derivatives
Objectives Find the slope of a secant line Find the slope of a tangent line Find the equation of a line tangent to a curve at a point Find the derivative of an equation
Slope of a line The slope of the line between points (a,f(a)) and (b,f(b)) of the function is:
Example 1 Slope between x = 3 and x =5 for the function: f(x) = x 2 – 4
Secant Line A line that goes through two points on a curve.
Example 2 Find an equation of the secant to: f(x) = x 2 – 4 when x = -1 and x = 3. Points: (-1,-3) and (3,5) Slope: Equation:
Generic Secant Line For any function f(x) find the slope of the secant line through: (x,f(x)) and (x+h,f(x+h) (x,f(x)) (x+h,f(x+h)) h
Generic Secant Line Points: (x,f(x)) and (x+h,f(x+h) (x,f(x)) (x+h,f(x+h)) Slope:
When the two points move very close together we have h->0. Write that limit. (x,f(x)) This is the slope of the tangent line – also known as the derivative
Example 3 Find the slope of the line tangent to f(x) = x + 1 at (1,2) Slope:
Example 4 Find the derivative of f(x) = x 2 Derivative:
Example 5 Find the derivative of: Derivative:
Example 6 Find the equation of the line tangent to f(x) = x 2 when x = 3 Slope = Derivative: Point on Curve: Equation:
Derivatives and graphs a b c d e Derivative Graph: a b c d e
Omit from Assignment: #4, 5, 8, 11, 14, 18, 19, 21
Unit 6 – Fundamentals of Calculus Section 6
Sec 3.1: Tangents and the Derivative at a Point
Remember: Derivative=Slope of the Tangent Line.
Equations of Tangent Lines
Slope and Equation of a line How to find the slop of a line? (x 1, y 1 ) (x 2, y 2 ) How to find the equation of a line? Sec 2.1: Rates of Change and.
Equation of a Tangent Line
The derivative as the slope of the tangent line (at a point)
Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x
Rate of change and tangent lines
Point Value : 20 Time limit : 2 min #1 Find. #1 Point Value : 30 Time limit : 2.5 min #2 Find.
The Derivative Chapter 3:. What is a derivative? A mathematical tool for studying the rate at which one quantity changes relative to another.
AP CALCULUS 1005: Secants and Tangents. Objectives SWBAT determine the tangent line by finding the limit of the secant lines of a function. SW use both.
Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.
THE DERIVATIVE AND THE TANGENT LINE PROBLEM
We will start from Chapter 2 Sections 2.1 to 2.8 MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag.
1.4 – Differentiation Using Limits of Difference Quotients
Implicit Differentiation. Objectives Students will be able to Calculate derivative of function defined implicitly. Determine the slope of the tangent.
Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.
3.8 Derivatives of Inverse Trigonometric Functions.
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